Discrete Approximation of Non-Compact Operators Describing Continuum-of-Alleles Models

Abstract

We consider the eigenvalue equation for the largest eigenvalue of certain kinds of non-compact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite rank, which constitutes a discretization procedure. For this purpose, two standard methods of approximation theory, the Nyström and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labelled by real numbers, commonly called continuum-of-alleles models.

A Nyström Method for the Eigenvalue Problem of a Class of Noncompact Operators

“…In [8], the necessary and sufficient conditions for existence and uniqueness of positive eigenfunctions of eigenvalue problem (4) are obtained. These conditions do not depend on the compactness of the interval [a, b] and remain the same even if the problem is defined on the real line.…”

“…Therefore, the necessary and sufficient conditions for uniqueness and existence of positive eigenfunctions of eigenvalue problem (4), obtained in [8], can be applied to the eigenvalue problem (1). The results are summarized in the following theorem.…”

“…Nevertheless, Redner [8] used some kinds of Nyström method for numerical approximation of the eigenvalue corresponding to the unique positive eigenfunction of a class of noncompact operators describing continuum-of-alleles (COA) model. The COA model occurs in population genetics, which is concerned with the (micro)evolution of the genetic composition of populations [3,8]. The eigenvalue problem studied by Redner is defined on the space L 1 ([a, b], ) as follows:…”

“…However, this similarity does not mean that the technique used by Redner in [8] in order to develop Nyström methods to the problem (4), can be Downloaded by [University of Tennessee, Knoxville] at 07:51 05 January 2015 applied directly to the problem (1). This is because the integral part in problem (1) is defined on the real line while the integral part in problem (4) is defined on the compact interval [a, b].…”

“…This is because the integral part in problem (1) is defined on the real line while the integral part in problem (4) is defined on the compact interval [a, b]. In this article, we develop the Nyström method described in [8], using the combination with the finite section method and implement it to numerical approximation of problem (1). The results can also be applied to the COA model (4) when defined on the real line.…”